Sampling and Reconstruction of Bandlimited Signals with Multi-Channel Time Encoding

Sampling is classically performed by recording the amplitude of the input at given time instants; however, sampling and reconstructing a signal using multiple devices in parallel becomes a more difficult problem to solve when the devices have an unknown shift in their clocks. Alternatively, one can record the times at which a signal (or its integral) crosses given thresholds. This can model integrate-and-fire neurons, for example, and has been studied by Lazar and Tóth under the name of "Time Encoding Machines". This sampling method is closer to what is found in nature. In this paper, we show that, when using time encoding machines, reconstruction from multiple channels has a more intuitive solution, and does not require the knowledge of the shifts between machines. We show that, if single-channel time encoding can sample and perfectly reconstruct a 2Ω-bandlimited signal, then M-channel time encoding can sample and perfectly reconstruct a signal with M times the bandwidth. Furthermore, we present an algorithm to perform this reconstruction and prove that it converges to the correct unique solution, in the noiseless case, without knowledge of the relative shifts between the machines. This is quite unlike classical multi-channel sampling, where unknown shifts between sampling devices pose a problem for perfect reconstruction.

Sampling at unknown locations: Uniqueness and reconstruction under constraints

Benjamin Bejar Haro, Golnooshsadat Elhami, Michalina Wanda Pacholska, Adam James Scholefield, Martin Vetterli

Traditional sampling results assume that the sample locations are known. Motivated by simultaneous localization and mapping (SLAM) and structure from motion (SfM), we investigate sampling at unknown locations. Without further constraints, the problem is often hopeless. For example, we recently showed that, for polynomial and bandlimited signals, it is possible to find two signals, arbitrarily far from each other, that fit the measurements. However, we also showed that this can be overcome by adding constraints to the sample positions. In this paper, we show that these constraints lead to a uniform sampling of a composite of functions. Furthermore, the formulation retains the key aspects of the SLAM and SfM problems, whilst providing uniqueness, in many cases. We demonstrate this by studying two simple examples of constrained sampling at unknown locations. In the first, we consider sampling a periodic bandlimited signal composite with an unknown linear function. We derive the sampling requirements for uniqueness and present an algorithm that recovers both the bandlimited signal and the linear warping. Furthermore, we prove that, when the requirements for uniqueness are not met, the cases of multiple solutions have measure zero. For our second example, we consider polynomials sampled such that the sampling positions are constrained by a rational function. We previously proved that, if a specific sampling requirement is met, uniqueness is achieved. In addition, we present an alternate minimization scheme for solving the resulting non-convex optimization problem. Finally, fully reproducible simulation results are provided to support our theoretical analysis.

2018

Uncertain Sampling with Certain Priors

Golnooshsadat Elhami

Sampling has always been at the heart of signal processing providing a bridge between the analogue world and discrete representations of it, as our ability to process data in continuous space is quite limited. Furthermore, sampling plays a key part in understanding how to efficiently capture, store and process signals. Shannon's sampling theorem states that if the original signal is known to have a limited bandwidth, we can retrieve the signal from uniformly-spaced samples, provided that the sampling rate is greater than twice the highest frequency in the signal. Here, we see two key attributes: prior knowledge on the original signal (limited bandwidth) and a constrained sampling setup (uniform samples at a particular rate). In this thesis, we make weaker assumptions on the sampling setup by assuming that some information, such as the sample positions, is lost. We show that under proper prior knowledge, we can reconstruct the signal from its samples uniquely or up to some equivalence class. We start by the problem of linear sampling of discrete signals, where the sample values are known, but their order is lost. In general, the original signal is impossible to retrieve from the samples, but we show that by taking out symmetry from the sampling vectors, we can reconstruct the signal uniquely. We provide an efficient algorithm to find the sample orders and thus reconstruct the original signal. We also study the problem of reconstructing a continuous signal from samples taken at unknown locations. The lost sample locations take away any hope of uniquely retrieving the signal without prior knowledge. We show that this problem is equivalent to reconstructing a composite of functions from uniformly spaced samples. Then we provide an efficient algorithm that can recover bandlimited signals warped by a linear function uniquely given enough sampling frequency. We then investigate a problem, dubbed shape from bandwidth, where we have uniform samples from a picture (projection) of an unknown surface that is painted with an unknown texture. The goal is to reconstruct the shape of the surface from these samples. We show that having prior knowledge of the texture bandwidth provides us with enough information to reconstruct the surface from its picture. We provide reconstruction algorithms for both orthogonal and central projections and provide equivalence classes of solutions in each case. Next, in two consecutive chapters, using techniques from geometrical signal processing, we offer new designs for 3-D barcodes, whose information can be retrieved from a single projection using penetrating waves from an unknown direction. Because of the unknown scan direction, the correct correspondence of the samples to the information bits in the barcodes is lost. In this case, we use the known shape of the barcode as prior knowledge to estimate the unknown scan direction from the samples, and then transform the reconstruction as a linear problem that can be solved efficiently. Finally, we cover the theory of coordinate difference matrices (CDMs): matrices that store mutual differences between coordinates of points (sensors, microphones, molecules, etc.) in space. We show how we can leverage specific properties of these matrices, such as their low rank, as prior knowledge in order to reconstruct the position of the points in space using CDMs. We use our reconstruction algorithm to solve many real-life signal processing problems.

EPFL2021

Shapes from Pixels

Mitra Fatemi

In today's digital world, sampling is at the heart of any signal acquisition device. Imaging devices are ubiquitous examples that capture two-dimensional visual signals and store them as the pixels of discrete images. The main concern is whether and how the pixels provide an exact or at least a fair representation of the original visual signal in the continuous domain. This motivates the design of exact reconstruction or approximation techniques for a target class of images. Such techniques benefit different imaging tasks such as super-resolution, deblurring and compression. This thesis focuses on the reconstruction of visual signals representing a shape over a background, from their samples. Shape images have only two intensity values. However, the filtering effect caused by the sampling kernel of imaging devices smooths out the sharp transitions in the image and results in samples with varied intensity levels. To trace back the shape boundaries, we need strategies to reconstruct the original bilevel image. But, abrupt intensity changes along the shape boundaries as well as diverse shape geometries make reconstruction of this class of signals very challenging. Curvelets and contourlets have been proved as efficient multiresolution representations for the class of shape images. This motivates the approximation of shape images in the aforementioned domains. In the first part of this thesis, we study generalized sampling and infinite-dimensional compressed sensing to approximate a signal in a domain that is known to provide a sparse or efficient representation for the signal, given its samples in a different domain. We show that the generalized sampling, due to its linearity, is incapable of generating good approximation of shape images from a limited number of samples. The infinite-dimensional compressed sensing is a more promising approach. However, the concept of random sampling in this scheme does not apply to the shape reconstruction problem. Next, we propose a sampling scheme for shape images with finite rate of innovation (FRI). More specifically, we model the shape boundaries as a subset of an algebraic curve with an implicit bivariate polynomial. We show that the image parameters are solutions of a set of linear equations with the coefficients being the image moments. We then replace conventional moments with more stable generalized moments that are adjusted to the given sampling kernel. This leads to successful reconstruction of shapes with moderate complexities from samples generated with realistic sampling kernels and in the presence of moderate noise levels. Our next contribution is a scheme for recovering shapes with smooth boundaries from a set of samples. The reconstructed image is constrained to regenerate the same samples (consistency) as well as forming a bilevel image. We initially formulate the problem by minimizing the shape perimeter over the set of consistent shapes. Next, we relax the non-convex shape constraint to transform the problem into minimizing the total variation over consistent non-negative-valued images. We introduce a requirement -called reducibility- that guarantees equivalence between the two problems. We illustrate that the reducibility effectively sets a requirement on the minimum sampling density. Finally, we study a relevant problem in the Boolean algebra: the Boolean compressed sensing. The problem is about recovering a sparse Boolean vector from a few collective binary tests. We study a formulation of this problem as a binary linear program, which is NP hard. To overcome the computational burden, we can relax the binary constraint on the variables and apply a rounding to the solution. We replace the rounding procedure with a randomized algorithm. We show that the proposed algorithm considerably improves the success rate with only a slight increase in the computational cost.